Methods, systems, apparatus and articles of manufacture to determine causal effects

ABSTRACT

Methods, systems, apparatus and articles of manufacture to determine causal effects are disclosed herein. An example apparatus includes a weighting engine to calculate a first set of weights corresponding to a first treatment dataset, a second set of weights corresponding to a second treatment dataset, and a third set of weights corresponding to a control dataset, the weighting engine to increase an operational efficiency of the apparatus by calculating the first set of weights, second set of weights, and third set of weights independently, a weighting response engine to calculate a first weighted response for the first treatment dataset, a second weighted response for the second treatment dataset, and determine a causal effect between the first treatment dataset and the second treatment dataset based on a difference between the first weighted response and the second weighted response, and a report generator to transmit a report to an audience measurement entity.

RELATED APPLICATION

This patent is a continuation of and claims priority to U.S. patentapplication Ser. No. 16/219,524, filed on Dec. 13, 2018, which claimsbenefit of U.S. Provisional Patent Application Ser. No. 62/685,741,filed on Jun. 15, 2018, and U.S. Provisional Patent Application Ser. No.62/686,499, filed on Jun. 18, 2018. U.S. patent application Ser. No.16/219,524, U.S. Provisional Patent Application Ser. No. 62/685,741, andU.S. Provisional Patent Application Ser. No. 62/686,499 are herebyincorporated herein by reference in their entireties. Priority to U.S.patent application Ser. No. 16/219,524, U.S. Provisional PatentApplication Ser. No. 62/685,741, and U.S. Provisional Patent ApplicationSer. No. 62/686,499 is hereby claimed.

FIELD OF THE DISCLOSURE

This disclosure relates generally to audience measurement and, moreparticularly, to methods, systems, apparatus and articles of manufactureto determine causal effects.

BACKGROUND

In recent years, market research efforts have collected market behaviorinformation to determine an effect of marketing campaign efforts. Duringsome marketing campaign efforts, adjustments are made to one or moremarket drivers, such as a promotional price of an item, an advertisementchannel (e.g., advertisements via radio, advertisements via television,etc.), and/or in-store displays. Market analysts attempt to identify adegree to which such adjustments to market drivers affect a marketingcampaign objective, such as increased unit sales.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example analysis engine to determine causaleffects for audience measurement in accordance with teachings disclosedherein.

FIGS. 2-5 are flowcharts representative of example machine readableinstructions that may be executed to implement the example analysisengine of FIG. 1.

FIG. 6 is a block diagram of an example processor platform structured toexecute the example machine readable instructions of FIGS. 2-5 toimplement the example analysis engine of FIG. 1.

Wherever possible, the same reference numbers will be used throughoutthe drawing(s) and accompanying written description to refer to the sameor like parts, elements, etc.

DETAILED DESCRIPTION

Market researchers seek to understand whether adjusting variables withintheir control group dataset have a desired effect. In some examples,variables that can be controlled by personnel and/or entities (e.g.,manufacturers, merchants, retailers, etc., generally referred to hereinas “market researchers”) interested in the desired effect include aprice of an item, a promotional price, a promotional duration, apromotional vehicle (e.g., an adjustment related to distributed mediasuch as television, radio, Internet, etc.), a package design, a feature,a quantity of ingredients, etc. In short, if the market researcher knowsthat changing a variable (e.g., a cause) leads to achievement of themarketing campaign objective (e.g., an effect), then similar marketingcampaigns can proceed with a similar expectation of success.

Industry standard statistical methodologies distinguish betweengathering data to identify a relationship between a variable (e.g., amarket driver under the control of the market researcher) and a result(e.g., an effect observed when the variable is present) versus whethersuch variables are the cause of the observed result. Stated differently,market researchers know that correlation does not necessarily meancausation. Positive correlations can be consistent with positive causaleffects, no causal effects, or negative causal effects. For example,taking cough medication is positively correlated with coughing, buthopefully has a negative causal effect on coughing.

Causation, unlike correlation, is a counterfactual claim in a statementabout what did not happen. The statement that “X caused Y” means that Yis present, but Y would not have been present if X were not present.Caution must be exercised by market researchers regarding potentialcompeting causes that may be present when trying to determine a cause ofobserved outcomes to avoid absurd conclusions. An example statementhighlighting such an absurd conclusion in view of causation is thatdriving without seat belts prevents deaths from smoking because it killssome people who would otherwise go on to die of smoking-related disease.Competing causes may be further illustrated in a statement from theNational Rifle Association that “guns don't kill people, people killpeople.” In particular, one competing cause is that if you take awayguns and you observe no deaths from gunshot wounds, then guns are acause. However, another competing cause is that if you take away peopleand you have no deaths from gunshot wounds, then people (e.g., shooters)are a cause. As such, both illustrate simultaneous causes of the sameoutcome. To frame an analysis in a manner that avoids extreme and/orotherwise absurd conclusions, the question of whether “X causes Y” isbetter framed as “how much does X affect Y.”

Determining causal effects from observation studies is a well-studiedproblem in the technical field of behavioral research. Determiningcausal effects from observation studies is different from experimentaldesigns as the subjects choose to be either in a treatment group or in acontrol group (in binary setting), in contrast to experimental designswhere the subjects are selected for a treatment group or a control groupclassification randomly. This non-random assignment of subjectsconfounds causal effects and adds bias (e.g., introduces error intofurther processing). To alleviate this bias, example algorithms such asPropensity Score matching and Inverse Propensity Score Weighting may beapplied. However, these algorithms only account for observable data ofthe treatment group dataset, which allows any hidden bias due to latentvariables to remain. Another downfall of such algorithms is that thealgorithms require large samples with substantial overlap between thetreatment group dataset and the control group dataset to produce resultswith some degree of reliability. Further, prior techniques, such asmultivariate reweighting, exhibit one or more data co-dependencies thatrequire that the treatment group dataset be analyzed first and thecontrol group dataset second. That is, prior techniques requireprocessing the treatment group dataset to process the control groupdataset. Such techniques require additional processing cycles whichburdens a processor.

Example methods, systems, apparatus and articles of manufacturedisclosed herein determine causal effects for audience measurementtechnologies and technical fields without the cultivation and/orprocessing of corresponding information that is typically required toremove bias during a causation study. Computational costs/burdensrequired to enable such technologies and/or behavioral fields are alsoreduced by examples disclosed herein by eliminating any need to acquire,sort, clean, randomize and/or otherwise manage separate information.Examples disclosed herein also reduce processing burdens whendetermining causal effects by avoiding and/or otherwise prohibitingcomputationally intensive parametric numerical approaches and/orregressions. Further, because examples disclosed herein avoid parametricnumerical approaches, causation determination results in a relativelylower error. Examples disclosed herein can perform operations withoutperforming individualized processing of treatment group datasets andcontrol group datasets (e.g., without requiring a processor to perform asubsequent weighting process of other computer-based causal effectprocesses such as multivariate reweighting.). That is, examplesdisclosed herein analyze the treatment group datasets and control groupdatasets independently (e.g., in a manner in which treatment groupdataset and control group dataset are not co-dependent) such thattreatment group dataset and control group dataset can be analyzedsubstantially simultaneously (e.g., utilizing the same processingcycle), which eliminates unnecessary processing cycles and reduces theburden on a processor.

FIG. 1 is a schematic illustration of an example environment 100constructed in accordance with the teachings of this disclosure todetermine causal effects for audience measurement. The illustratedexample of FIG. 1 includes an example analysis engine 102 and one ormore treatment/control data store(s) 104. The example analysis engine102 includes an example treatment/control data interface 106communicatively coupled to the one or more treatment/control datastore(s) 104, an example covariate engine 108, an example weightingengine 110, an example weighting response engine 112, and an examplereport generator 114. In some examples, the covariate engine 108 is ameans for generating a covariate, or a covariate generating means. Insome examples, the weighting engine 110 is a means for calculating aweight, or a weight calculating means. In some examples, the weightingresponse engine 112 is a means for determining a weighted response, or aweighted response determining means. In some examples, the reportgenerator 114 is a means for generating a report, or a report generatingmeans.

In operation, the example treatment/control data interface 106 of FIG. 1retrieves and/or otherwise receives treatment/control group datasetsfrom the example treatment/control data store(s) 104. Example methods,apparatus, systems and/or articles of manufacture disclosed hereindiscuss treatment/control group datasets related to an advertisementcampaign, but examples disclosed herein are not limited thereto. Otherexample treatment/control group datasets for which causal effects may bedetermined include, but are not limited to, drug trial data, tweets,product purchase instances, etc. The treatment/control group datasets tobe described in connection with example operation of the analysis engine102 includes data for males and females that (a) were exposed to anadvertisement of interest and (b) were not exposed to the advertisementof interest. In some examples, the treatment/control group datasets mayinclude an effect value that relates to an amount of change or perceivedchange when either viewing an advertisement of interest or an amount ofchange or perceived change when not viewing the advertisement ofinterest. Additionally, the treatment/control group datasets may includeother demographic and/or categorical information that may be of interestsuch as age, race, economic class, etc., for example.

The example covariate engine 108 of FIG. 1 generates data categoriesbased on the information from the treatment/control data store(s) 104.For example, the covariate engine 108 identifies an individual (i) fromthe treatment/control group datasets and identifies weights and/orcovariates (e.g., demographics, age, race, etc.) associated with theindividual for processing. In some examples, the treatment/controldatasets may not include weights (e.g., the treatment/control datasetsonly include covariates). The example covariate engine 108 separates thetreatment/control group datasets such that they are mutually exclusive(e.g., the treatment group dataset is separate from the control groupdataset). Once the data has been separated, the example covariate engine108 maximizes entropy (H) subject to the weights (w) summing to 100%such that the weights on the control group dataset match the sampleaverage of the weights in the treatment group dataset. In some examples,the covariate engine 108 maximizes the entropy in a manner consistentwith example Equation 1.

$\begin{matrix}{{{Maximize}\mspace{14mu} w}{H = {- {\sum\limits_{Z_{i} = 0}{w_{i}{\log \left( w_{i} \right)}}}}}{{Subject}\mspace{14mu} {to}}{{\sum\limits_{Z_{i} = 0}w_{i}} = 1}{{\sum\limits_{Z_{i} = 0}{w_{i}{c_{j}\left( X_{i} \right)}}} = {\frac{1}{n_{1}}{\sum\limits_{Z_{i} = 1}{c_{j}\left( X_{i} \right)}}}}{{j = 1},\ldots \mspace{20mu},m}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

In the illustrated example of Equation 1, w represents the weight for anindividual person (i) in the control, c represents the covariate (j)currently being processed (e.g., demographic, age, income, etc.), Xrepresents all of the covariates for that individual, n₁ represents thenumber of people in the observed treatment, and Z represents what wasobserved for that individual. For example, Z=0 represents the individualwas part of the control group dataset, and Z=1 represents the individualwas part of the treatment group dataset. The covariate engine 108maximizes the entropy to mitigate illogical results. In effect, suchentropy maximization efforts by the example covariate engine 108 reducesand/or otherwise minimizes a bias effect.

In some examples, the covariate engine 108 utilizes a weighted averageΣ_(z) _(i) ₌₀w_(i) ^(EB)Y_(i) to estimate a counterfactual meanE[Y(0)|T=1], where E represents the expected value of all individuals orsubjects in both the treatment group dataset and the control groupdataset, Y(0) represents a response of not viewing the advertisement,and T=1 is the treatment indicator (e.g., T=0 represents controlindicator). That is, E[Y(0)|T=1] represents a counterfactual response ofnot viewing the advertisement, but the individual actually viewed theadvertisement (e.g., an individual in the treatment viewed anadvertisement, but was classified as not viewing the advertisement). Insome examples, the covariate engine 108 determines the average treatmenteffect for the treated (ATT) represented by {circumflex over (γ)}_(ATT)^(EB), where EB stands for Entropy Balanced, in a manner consistent withexample Equation 2. The example covariate engine 108 also maximizes theEntropy (H) for the control group dataset in a manner consistent withexample Equation 3, where v represents the weights for the control groupdataset.

$\begin{matrix}{{\hat{\gamma}}_{ATT}^{EB} = {{\sum\limits_{Z_{i} = 1}\frac{1}{n_{1}}} - {\sum\limits_{Z_{i} = 0}{w_{i}^{EB}Y_{i}}}}} & {{Equation}\mspace{14mu} 2} \\{{{Maximize}\mspace{14mu} v}{H = {- {\sum\limits_{Z_{i} = 1}{v_{i}{\log \left( v_{i} \right)}}}}}{{Subject}\mspace{14mu} {to}}{{\sum\limits_{Z_{i} = 1}v_{i}} = 1}{{\sum\limits_{Z_{i} = 1}{v_{i}{c_{j}\left( X_{i} \right)}}} = {\frac{1}{n_{0}}{\sum\limits_{Z_{i} = 0}{c_{j}\left( X_{i} \right)}}}}{{j = 1},\ldots \mspace{14mu},m}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

The example covariate engine 108 estimates the Average Treatment effectfor the control (ATC) in a manner consistent with example Equation 4.

$\begin{matrix}{{\hat{\gamma}}_{ATC}^{EB} = {{\sum\limits_{z_{i} = 1}{v_{i}^{EB}Y_{i}}} - {\sum\limits_{z_{i} = 0}{\frac{1}{n_{0}}Y_{i}}}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

When determining ATT and ATC, the constraints must match to the sampleaverage of the other group. For example, the constraints for thetreatment group dataset must match the sample average of the controlgroup dataset. However, for ATE, both sets of weights must vary. Thatis, the overlap is needed because it would be mathematically impossiblefor the constraints of the treatment group dataset to match the sampleaverage of the control group dataset, for example, if no overlap waspresent. For example, assuming two groups of ages: group 1 being ages20-30 and group 2 being ages 80-90, it is impossible to match group 1 togroup 2 because there is no overlap. However, assuming another twogroups of ages: group 1 being ages 20-30 and group 2 being ages 15-40, amatch can be made to a same average value (e.g., 27).

In some examples, the covariate engine 108 of FIG. 1 determines a firstset of covariates to be processed for a treatment group dataset of anadvertisement campaign and a second set of covariates to be processedfor a control group dataset of the advertisement campaign. In someexamples, the treatment group dataset is indicative of participant (orindividual or subject) behaviors associated with exposure to anadvertisement, while the control group dataset is indicative ofparticipant behaviors associated with not being exposed to theadvertisement. In some examples, the covariate engine 108 processes thetreatment/control group datasets from the example treatment/control datastore(s) 104 to determine which covariates to balance. For example, thecovariate engine 108 may separate the data into treatment group datasetsand control group datasets. The covariate engine 108 may identify allthe covariates present in the treatment group datasets and all thecovariates present in the control group datasets, for example. Theexample covariate engine 108 identifies covariates that are similar(e.g., include at least one of the same covariate) across both thetreatment group datasets and control group datasets. The examplecovariate engine 108 may identify these covariates as covariates tobalance.

The example weighting engine 110 of FIG. 1 calculates a first set ofweights for the first set of covariates corresponding to the treatmentgroup dataset, and calculates a second set of weights for the second setof covariates corresponding to the control group dataset using maximumentropy. For example, the example weighting engine 110 may apply themaximum entropy techniques in a manner consistent with example Equation1 described above, or in a manner consistent with example Equations 5and 6 described in more detail below, to find weights for eachparticipant in the control group dataset and each participant in thetreatment group dataset so the weighted covariates are equal betweenboth groups. In some examples, the first set of weights is to equal thesecond set of weights. However, the example weighting engine 110determines the weights for each of the treatment group dataset andcontrol group dataset simultaneously. As used herein, “simultaneously”refers to at substantially the same time (e.g., a same clock cycle)and/or calculated without inter-dependence on the other group. As such,the weighting of the treatment group dataset neither influences nor isdependent upon the weighting of the control group dataset. Suchweighting by the example weighting engine 110 reduces the amount ofprocessing cycles required to determine weights for the respectivecontrol and/or treatment group datasets because a processor does notneed to execute a first processing cycle (e.g., a group of firstprocessing cycles) for the treatment group dataset and a secondprocessing cycle (e.g., a group of second processing cycles) for thecontrol group dataset. Instead, the first processing cycle is sufficientwith regard to examples disclosed herein. Further, the weighting engine110 increases the operational efficiency of the analysis engine 102 bysimultaneously determining the weights for both the treatment groupdataset and the control group dataset.

The example weighting response engine 112 calculates a weighted responsevalue for the treatment group dataset and a weighted response value forthe control group dataset based on the first set of weights and thesecond set of weights without requiring a processor to perform asubsequent weighting process of other computer-based causal effectprocesses. That is, the weighting response engine 112 does not need afirst processing cycle to determine a weighted response for thetreatment group dataset, and a second processing cycle to determine theweighted response for the control group dataset. For example, theweighting response engine 112 bypasses multivariate reweighting bycalculating a weighted response for the datasets based on the equationsdescribed in more detail below. The example weighting response engine112 outputs a weighted treatment group dataset measurement that is on acommon scale (e.g., a same unit of measure, a compatible unit ofmeasure, etc.) with the weighted control group dataset. As such, theexample weighting response engine 112 outputs common scale measurementsthat can be utilized in subsequent processing. The example weightingresponse engine 112 determines a difference between the weightedtreatment group dataset and the weighted control group dataset todetermine the Average Treatment Effect (ATE) as described in more detailbelow. The resulting ATE from the weighting response engine 112represents the potential outcome of an individual viewing anadvertisement of interest. For example, the resulting ATE may representthe increase in sales per person based on an individual viewing anadvertisement. In some examples, an ATE measurement may be calculatedfor multiple advertisements, which are subsequently compared to identifythe most effective advertisement.

The example report generator 114 generates a report indicating a causaleffect of the advertisement campaign based on a difference between theweighted response for the treatment group dataset and the weightedresponse for the control group dataset. For example, the example reportgenerator 114 receives and/or retrieves the ATE results from theweighting response engine 112, and generates a report. The reportgenerated by the example report generator 114 may subsequently beprovided to a measurement entity and/or another interested party. Insome examples, the report generator 114 displays the report on a devicevia a webpage in a first state with a set of options. For example, thereport generator 114 may display the report on a device via a webpage ina first state which displays the treatment group data set and controlgroup dataset with selectable drop down options (e.g., different levelsof detail such as group level, individual level, etc.) which change thestate of the display (e.g., selecting and option of the drop down menufor the treatment group dataset changes the state of the display to asecond state based on the option selected). The example set of optionsmay be selectable by a user to change the state of the display to viewdifferent types of information in the report.

The following equations may be utilized by the covariate engine 108, theweighting engine 110, the weighting response engine 112 and/or moregenerally the analysis engine 102 to determine the causal effect foraudience measurement. In some examples, the example analysis engine 102utilizes Equation 1 to find weights on the covariate of the controlgroup dataset to match the expected value (or sample average) of thecovariate in the treatment group dataset. The example analysis engine102 matches the covariate moments of the control group dataset, subjectto them having to equal the treatment group dataset. The justificationfor this is typically a counterfactual quantity must be estimated byweighting the counterfactual quantity and matching the counterfactualquantity to treatment group dataset covariates. However, for ATC it isthe reverse, the example analysis engine 102 determines weights on thetreatment group dataset to match the control group dataset in a mannerconsistent with example Equation 3.

Examples disclosed herein allow both sets of weights (e.g., weights ofthe treatment group dataset, weights of the control group dataset) tovary. The example analysis engine 102 weights both the control groupdataset and treatment group dataset to estimate an average treatmenteffect (ATE). The example analysis engine 102 utilizes a double entropybalancing where two sets of weights (e.g., treatment group datasetweights, control group dataset weights) are simultaneously solved suchthat the resulting weights of the treatment group dataset and theresulting weights of the control group dataset match a weighted average.The example weighting engine 110 maximizes an entropy value (H) subjectto particular linear constraints, and for the resulting entropy value tobe consistent with logic, the example weighting engine 110 utilizes ageneralized KL-divergence value, of which Maximum Entropy is a specialcase. In some examples, the weighting engine 110 maximizes an entropyvalue in a manner consistent with Equation 5.

$\begin{matrix}{{{{Maximize}\mspace{14mu} w},v}{{H = {\left( {- {\sum\limits_{Z_{i} = 1}{v_{i}{\log \left( v_{i} \right)}}}} \right) + \left( {- {\sum\limits_{Z_{i} = 0}{w_{i}{\log \left( w_{i} \right)}}}} \right)}}{Subject}\mspace{14mu} {to}\mspace{14mu} {the}\mspace{14mu} {following}\mspace{14mu} {linear}\mspace{14mu} {constraints}}{{\sum\limits_{Z_{i} = 1}v_{k}} = 1}{{\sum\limits_{Z_{i} = 0}w_{k}} = 1}{{\sum\limits_{Z_{i} = 1}{v_{i}{c_{j}\left( X_{i} \right)}}} = {\sum\limits_{Z_{i} = 0}{w_{i}{c_{j}\left( X_{i} \right)}}}}{{j = 1},\ldots \mspace{14mu},m}} & {{Equation}\mspace{14mu} 5}\end{matrix}$

In the illustrated example of Equation 5, (k) represents a group (e.g.,treatment group dataset 1, control group dataset 4, etc.). The exampleweighting engine 112 determines the Average Treatment Effect in a mannerconsistent with Equation 6.

$\begin{matrix}{{\hat{\gamma}\; {ATE}} = {{\sum\limits_{Z_{i} = 1}{v_{i}Y_{i}}} - {\sum\limits_{Z_{i} = 0}{w_{i}Y_{i}}}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

In the illustrated example of Equation 6, Y represents an outcome (e.g.,monetary increase) of viewing an advertisement which comes from thecontrol data. The resulting ATE value is representative of a potentialdollar increase if the advertisement of interest (e.g., theadvertisement analyzed using Equations 5 and 6) is viewed by anindividual. Solving for the weighted propensity score using the weightssolved for using Equations 5 and 6, and the covariates which are asubset of the covariates used in solving for those weights, theresulting probability of being in the treatment group dataset would be50% for every individual in the entire sample. With the weights solvedusing Equations 5 and 6, the covariates are independent of assignment oftreatment group dataset and it is a fair coin toss (e.g., random) ofbeing assigned to the treatment group dataset. Although an unbiased coinis not required for causal estimates, just random assignment, thestatistical power of the results are maximized and/or otherwise improvedwhen the assignment is unbiased and 50%:50% for a binary classificationof treatment group dataset and control group dataset.

Solving for the constraints when they are equal may seem odd becausethey are unknown. However, this can be done by rephrasing the problem,which is mathematically identical. The constraint that the sum must be100% is usually explicitly incorporated within as a normalization, whichis unfortunate as it treats the constraint unique over any otherconstraint. Treating the sum=100% like any other constraint results inm+2 constraining equations among n=n₀+n₁ probabilities. As Entropy isadditive this also does not change the optimization function. In otherwords, by changing notation, the entropy maximization is simplified in amanner consistent with example Equation 7.

The analysis engine 102 allows the weights to vary (e.g., the weightsare calculated independent of one another). As such, the weightingengine 110 may change the notation of Equation 5 in a manner consistentwith Equation 7.

$\begin{matrix}{{{Maximize}\mspace{14mu} p}{H = {- {\sum\limits_{i = 1}^{n_{1} + n_{0}}{p_{i}{\log \left( p_{i} \right)}}}}}{{Subject}\mspace{14mu} {to}}{{\sum\limits_{i = 1}^{n_{1}}p_{i}} = 1}{{\sum\limits_{i = {n_{1} + 1}}^{n}p_{i}} = 1}{{{\sum\limits_{Z_{i} = 1}{p_{i}{c_{j}\left( X_{i} \right)}}} - {\sum\limits_{Z_{i} = 0}{p_{i}{c_{j}\left( X_{i} \right)}}}} = 0}{{j = 1},\ldots \mspace{14mu},m}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

In the illustrated example of Equation 7, p represents both the weightsfor the treatment group dataset and the weights for the control groupdataset. That is, both the weights for the treatment group dataset andthe weights for the control group dataset are mapped to a uniformweighting identifier (p). Both normalizations are incorporated as twoseparate constraints and result in the additional m covariate balancingconstraints. In the illustrated example of Equation 7, because bothweights are represented by p, the weighting engine 110 keeps track ofthe treatment group dataset weights and the control group datasetweights by utilizing positive and negative Lagrange Multipliers (λ) in amanner consistent with Equation 8.

$\begin{matrix}{{v_{i} = {\exp \left( {\lambda_{0}^{(1)} + {\sum\limits_{j = 1}^{m}{\lambda_{j}{c_{j}\left( x_{i} \right)}}}} \right)}}{w_{i} = {\exp \left( {\lambda_{0}^{(0)} + {\sum\limits_{j = 1}^{m}{\left( {- \lambda_{j}} \right){c_{j}\left( x_{i} \right)}}}} \right)}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

In the illustrated example of Equation 8 the Lagrange Multipliers (λ)are unknown, and as discussed above, v represents weights correspondingto the control group dataset, and w represents weights corresponding tothe treatment group dataset. For example, assuming three subjects in thecontrol group dataset and two in the treatment group dataset, theweighting engine 110 balances the weights across two covariates. Thelabels of the covariates shown in example Table 1 are arbitrary andsmall numbers used only for illustration.

TABLE 1 Treated Control Outcome Outcome Index Covariates (Y1) Covariates(Y0) 1 [4,8] 7 [2,3] 2 2 [7,1] 5 [8,5] 4 3 [4,1] 1

To solve for the Lagrange Multipliers (A), the weighting engine 110maximizes Equation 7 subject to the following example constraint matrixshown in example Equation 9.

$\begin{matrix}{{\begin{bmatrix}1 & 1 & 0 & 0 & 0 \\0 & 0 & 1 & 1 & 1 \\4 & 7 & {- 2} & {- 8} & {- 4} \\8 & 1 & {- 3} & {- 5} & {- 1}\end{bmatrix}\begin{bmatrix}p_{1} \\p_{2} \\p_{2} \\p_{4} \\p_{5}\end{bmatrix}} = \begin{bmatrix}1 \\1 \\0 \\0\end{bmatrix}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

In the illustrated example of Equation 9, elements (p₁, p₂) map to thetreatment group dataset weights (v₁, v₁), and elements (p₃, p₄, p₅) mapto the control group dataset weights (w₁, w₃, w₃). The analysis engine102 inputs the resulting variables into Equation 8 in a mannerconsistent with example Equation 10.

v ₁=exp(λ₀ ⁽¹⁾+(λ₁)4+(λ₂)8)

v ₂=exp(λ₀ ⁽¹⁾(1)+(λ₁)7+(λ₂)1)

w ₁=exp(λ₀ ⁽⁰⁾(1)+(−λ₁)2+(−λ₂)3)

w ₂=exp(λ₀ ⁽⁰⁾(1)+(−λ₁)8+(−λ₂)5)

w ₃=exp(λ₀ ⁽⁰⁾(1)+(−λ₁)4+(−λ₂)1)  Equation 10:

The example weighting engine 110 solves Equation 10 to determine theLagrange Multipliers. The example weighting engine 110 substitutes theresulting Lagrange Multipliers into Equation 10 to produce the resultingweighted covariates in a manner consistent with example Equation 11. Assuch, the weighted covariates for the control group dataset are equal tothe weighted covariates for the treatment group dataset.

$\begin{matrix}{{v_{1} = {{0.3}861}}{v_{2} = 0.6139}\overset{\_}{w_{1} = {{0.2}152}}{w_{2} = {{0.5}680}}{w_{3} = {{0.2}168}}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

The example weighting response engine 112 inputs the resulting weightsfor the treatment and control group datasets into Equation 6 todetermine the Average Treatment Effect in a manner consistent withEquation 12.

$\begin{matrix}\begin{matrix}{{\hat{\gamma}\; {ATE}} = {{\sum\limits_{Z_{i} = 1}{v_{i}Y_{i}}} - {\sum\limits_{Z_{i} = 0}{w_{i}Y_{i}}}}} \\{= {\left( {{(0.3861)(7)} + {(0.6139)(5)}} \right) -}} \\{\left( {{(0.2152)(2)} + {(0.5680)(4)} + {(0.2168)(1)}} \right)} \\{= {5.7722 - 2.9193}} \\{= 2.8528}\end{matrix} & {{Equation}\mspace{14mu} 12}\end{matrix}$

As such, the resulting ATE of 2.8528 represents a potential increase of$2.8528 for viewing an advertisement of interest.

Examples disclosed herein utilize the same principle to balancecovariates simultaneously across multiple treatment group datasetlevels. In the section above, the analysis engine 102 symbolicallyconstrains A−B=0, where A and B represent weighted covariates among thetreatment group dataset and control group dataset, and the differencemust be zero and therefore A=B. To incorporate a third group (e.g.,group C) the analysis engine 102 adds another set of constraintsrelating group B with group C, such that B−C=0. As such, A=B and B=C,and by transitive relation of equality A=C, A=B=C.

The example covariate engine 108 can therefore solve for multiple groupsby linking each new group with the previous, A−B=0, B−C=0, C−D=0, D−E=0,etc.

For example, example Table 2 represents a control group dataset and twotreatment group dataset levels.

TABLE 2 Second Level Treatment index Covariates Outcome (Y²) 1 [10, 0] 9 2 [1, 1] 5 3 [8, 4] 10 4 [9, 3] 2 First Level Treatment indexCovariates Outcome (Y¹) 1 [4, 8] 7 2 [7, 1] 5 Control index CovariatesOutcome (Y⁶) 1 [2, 3] 2 2 [8, 5] 4 3 [4, 1] 1

The example weighting engine 110 maximizes entropy value (H) subject toparticular linear constraints, similar to Equation 7 above. In someexamples, the weighting engine 110 maximizes an entropy value in amanner consistent with Equation 13 when multiple groups (e.g., twotreatment group dataset levels and three control group dataset levels, 4treatment group dataset level and 1 control group dataset, etc.) arebeing analyzed.

$\begin{matrix}{{{Maximize}\mspace{14mu} p}{H = {- {\sum\limits_{i = 1}^{n_{2} + n_{1} + n_{0}}{p_{i}{\log \left( p_{i} \right)}}}}}{{Subject}\mspace{14mu} {to}}{{\sum\limits_{i = 1}^{n_{2}}p_{i}} = 1}{{\sum\limits_{i = {n_{2} + 1}}^{n_{2} + n_{1}}p_{i}} = 1}{{\sum\limits_{i = {n_{2} + n_{1} + 1}}^{n_{2} + n_{1} + n_{0}}p_{i}} = 1}{{{\sum\limits_{Z_{i} = 2}{p_{i}{c_{j}\left( X_{i} \right)}}} - {\sum\limits_{Z_{i} = 1}{p_{i}{c_{j}\left( X_{i} \right)}}}} = 0}{{j = 1},\ldots \mspace{14mu},m}{{{\sum\limits_{Z_{i} = 1}{p_{i}{c_{j}\left( X_{i} \right)}}} - {\sum\limits_{Z_{i} = 0}{p_{i}{c_{j}\left( X_{i} \right)}}}} = 0}{{j = 1},\ldots \mspace{14mu},{m\left( {{{{\sum\limits_{Z_{i} = 0}{p_{i}{c_{j}\left( X_{i} \right)}}} - {\sum\limits_{Z_{i} = 2}{p_{i}{c_{j}\left( X_{i} \right)}}}} = {{0\mspace{14mu} j} = 1}},\ldots \mspace{14mu},m} \right)}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

In the illustrated example of Equation 13, the last group of equalityconstraints are not logically needed, but are incorporated for purposesof illustration. The weighting engine 110 maximizes Equation 13 subjectto the following constraint matrix in example Equation 14.

$\begin{matrix}{{\begin{bmatrix}1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\10 & 1 & 8 & 9 & {- 4} & {- 7} & 0 & 0 & 0 \\9 & 1 & 4 & 3 & {- 8} & {- 1} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 4 & 7 & {- 2} & {- 8} & {- 4} \\0 & 0 & 0 & 0 & 8 & 1 & {- 3} & {- 5} & {- 1} \\{- 10} & {- 1} & {- 8} & {- 9} & 0 & 0 & 2 & 8 & 4 \\{- 9} & {- 1} & {- 4} & {- 3} & 0 & 0 & 3 & 5 & 1\end{bmatrix}\begin{bmatrix}p_{1} \\p_{2} \\p_{2} \\p_{4} \\p_{5} \\p_{6} \\p_{7} \\p_{8} \\p_{9}\end{bmatrix}} = \begin{bmatrix}1 \\1 \\1 \\0 \\0 \\0 \\0 \\0 \\0\end{bmatrix}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

In the illustrated example of Equation 14, elements (p₁, p₂, p₃, p₄) mapto the second level treatment group dataset weights (w₁ ⁽²⁾w₂ ⁽²⁾w₃⁽²⁾w₄ ⁽²⁾), elements (p₅, p₆) map to the first level treatment groupdataset weights(w₁ ⁽¹⁾w₂ ⁽¹⁾), and elements (p₇, p₈, p₉) map to thecontrol group dataset weights (w₁ ⁽⁰⁾w₂ ⁽⁰⁾w₃ ⁽⁰⁾). The exampleweighting engine 110 subsequently maps each constraint with a LagrangeMultiplier, as illustrated in example Equation 15.

$\begin{matrix}\left. \begin{bmatrix}1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\10 & 1 & 8 & 9 & {- 4} & {- 7} & 0 & 0 & 0 \\9 & 1 & 4 & 3 & {- 8} & {- 1} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 4 & 7 & {- 2} & {- 8} & {- 4} \\0 & 0 & 0 & 0 & 8 & 1 & {- 3} & {- 5} & {- 1} \\{- 10} & {- 1} & {- 8} & {- 9} & 0 & 0 & 2 & 8 & 4 \\{- 9} & {- 1} & {- 4} & {- 3} & 0 & 0 & 3 & 5 & 1\end{bmatrix}\Rightarrow\begin{bmatrix}\lambda_{0}^{(2)} \\\lambda_{0}^{(1)} \\\frac{\lambda_{0}^{(0)}}{\lambda_{1}^{(2)}} \\\frac{\lambda_{2}^{(2)}}{\lambda_{1}^{(1)}} \\\frac{\lambda_{2}^{(1)}}{\lambda_{1}^{(0)}} \\\lambda_{2}^{(0)}\end{bmatrix} \right. & {{Equation}\mspace{14mu} 15}\end{matrix}$

The weighting engine 110 solves Equation 15 to determine the resultingweighted covariates. The example weighting engine 110 substitutesLagrange Multipliers into Equation 8 to produce the resulting weightedcovariates in a manner consistent with example Equation 16.

$\begin{matrix}{{w_{1}^{(2)} = {\exp \left( {{\lambda_{0}^{(2)}(1)} + {\left( {\lambda_{1}^{(2)} - \lambda_{1}^{(0)}} \right)\left( {10} \right)} + {\left( {\lambda_{2}^{(2)} - \lambda_{2}^{(0)}} \right)(9)}} \right)}}{w_{2}^{(2)} = {\exp \left( {{\lambda_{0}^{(2)}(1)} + {\left( {\lambda_{1}^{(2)} - \lambda_{1}^{(0)}} \right)(1)} + {\left( {\lambda_{2}^{(2)} - \lambda_{2}^{(0)}} \right)(1)}} \right)}}w_{3}^{(2)} = {\exp \left( {{\lambda_{0}^{(2)}(1)} + {\left( {\lambda_{1}^{(2)} - \lambda_{1}^{(0)}} \right)(8)} + {\left( {\lambda_{2}^{(2)} - \lambda_{2}^{(0)}} \right)(4)}} \right)}} & {{Equation}\mspace{14mu} 16} \\{w_{4}^{(2)} = {\exp \left( {{\lambda_{0}^{(2)}(1)} + {\left( {\lambda_{1}^{(2)} - \lambda_{1}^{(0)}} \right)(9)} + {\left( {\lambda_{2}^{(2)} - \lambda_{2}^{(0)}} \right)(3)}} \right)}} & \; \\\overset{\_}{w_{1}^{(1)} = {\exp \left( {{\lambda_{0}^{(1)}(1)} + {\left( {\lambda_{1}^{(1)} - \lambda_{1}^{(2)}} \right)(4)} + {\left( {\lambda_{2}^{(1)} - \lambda_{2}^{(2)}} \right)(8)}} \right)}} & \; \\{{w_{2}^{(1)} = {\exp \left( {{\lambda_{0}^{(1)}(1)} + {\left( {\lambda_{1}^{(1)} - \lambda_{1}^{(2)}} \right)(7)} + {\left( {\lambda_{2}^{(1)} - \lambda_{2}^{(2)}} \right)(1)}} \right)}}\overset{\_}{w_{1}^{(0)} = {\exp \left( {{\lambda_{0}^{(0)}(1)} + {\left( {\lambda_{1}^{(0)} - \lambda_{1}^{(1)}} \right)(2)} + {\left( {\lambda_{2}^{(0)} - \lambda_{2}^{(1)}} \right)(3)}} \right)}}} & \; \\{{w_{2}^{(0)} = {\exp \left( {{\lambda_{0}^{(0)}(1)} + {\left( {\lambda_{1}^{(0)} - \lambda_{1}^{(1)}} \right)(8)} + {\left( {\lambda_{2}^{(0)} - \lambda_{2}^{(1)}} \right)(5)}} \right)}}{w_{3}^{(0)} = {\exp \left( {{\lambda_{0}^{(0)}(1)} + {\left( {\lambda_{1}^{(0)} - \lambda_{1}^{(1)}} \right)(4)} + {\left( {\lambda_{2}^{(0)} - \lambda_{2}^{(1)}} \right)(1)}} \right)}}} & \;\end{matrix}$

The example weighting engine 110 solves Equation 16 to determine theLagrange Multipliers: λ₀ ⁽²⁾=−0.8701, λ₀ ⁽¹⁾=0.0003, λ₀ ⁽⁰⁾=−2.1164, λ₁⁽²⁾=−0.0141, λ₂ ⁽²⁾=0.0406, λ₁ ⁽¹⁾=−0.0690, λ₂ ⁽¹⁾=−0.0531, λ₁⁽⁰⁾=0.0831, λ₂ ⁽⁰⁾=0.0126. The example weighting engine 110 substitutesthe resulting Lagrange Multipliers into Equation 16 to produce theresulting weighted covariates in a manner consistent with exampleEquation 17. As such, the weighted covariates for all three groups arenow equal to each other.

$\begin{matrix}{{{w_{1}^{(2)} = {{0.2}039}}{w_{2}^{(2)} = {{0.3}909}}{w_{3}^{(2)} = {{0.2}153}}}\frac{w_{4}^{(2)} = {{0.1}899}}{w_{1}^{(1)} = {{0.3}795}}\frac{w_{2}^{(1)} = 0.6205}{w_{1}^{(0)} = {0.1989}}{w_{2}^{(0)} = {{0.5}648}}{w_{3}^{(0)} = {{0.2}364}}} & {{Equation}\mspace{14mu} 17}\end{matrix}$

The example weighting response engine 112 inputs the resulting weightsfor the second treatment group datasets into Equation 18 to determinethe ATE for the second treatment group dataset, and inputs the resultingweights for the first treatment group dataset into Equation 19 todetermine the ATE for the first treatment group dataset.

$\begin{matrix}\begin{matrix}{{\hat{\gamma}}_{ATE}^{(2)} = {{\sum\limits_{Z_{i} = 2}{w_{i}^{(2)}Y_{i}^{(2)}}} - {\sum\limits_{Z_{i} = 0}{w_{i}^{(0)}Y_{i}^{(0)}}}}} \\{= {{6.3222} - {{2.8}932}}} \\{= 3.4290}\end{matrix} & {{Equation}\mspace{14mu} 18} \\\begin{matrix}{{\hat{\gamma}}_{ATE}^{(1)} = {{\sum\limits_{Z_{i} = 1}{w_{i}^{(1)}Y_{i}^{(1)}}} - {\sum\limits_{Z_{i} = 0}{w_{i}^{(0)}Y_{i}^{(0)}}}}} \\{= {5.7591 - {{2.8}932}}} \\{= 2.8659}\end{matrix} & {{Equation}\mspace{14mu} 19}\end{matrix}$

As such, the resulting ATE of 3.4290 for the second treatment grouprepresents a potential increase of $3.4290 for viewing the advertisementof the second control group dataset, and the resulting ATE of 2.8659 forthe first treatment group dataset represents a potential increase of$2.8659 for viewing the advertisement of the first control groupdataset. Thus, the report generator 114 may generate a report indicatingthe resulting potential increases for each advertisement.

In some examples, assuming K groups, k=1, . . . , K, with each grouphaving n_(k) individuals, i=1, . . . , n_(k). Additionally, there are Jcovariates to balance across all K groups (e.g., these may be firstmoments, second moments, or any other combination). As used herein,“moments” refer to covariates (e.g., demographic characteristics) ofindividuals in a group dataset, where the nth moment indicates thecovariate taken to the nth power (e.g., first moment represented byc(x)=x, second moment represented by c(x)=x². As such, the post-weightedzero moments sum to 100%, and the post-weighted first moments areutilized for covariate balancing (e.g., weighted average) as disclosedherein. In some examples, the second moment may have a square of thecovariates that are also balanced. This is beneficial as the balancingof both the first moments and the second moments results in weightedmeans that sum to 100%, and variances that sum to 100%. The exampleweighting engine 110 does not specify what the weighted covariate shouldbe, but maximizes an entropy value (H) subject to particular linearconstraints in a manner consistent with Equation 20

$\begin{matrix}{{Maximize}{H = {\sum\limits_{k + 1}^{K}\left( {- {\sum\limits_{i = 1}^{n_{k}}{w_{i}^{(k)}{\log \left( w_{i}^{(k)} \right)}}}} \right)}}{{Subject}\mspace{14mu} {to}\mspace{14mu} {the}\mspace{14mu} {following}\mspace{14mu} {constraints}}{{\sum\limits_{i = 1}^{n_{k}}w_{i}^{(k)}} = 1}{{{\forall k} = 1},\ldots \mspace{14mu},K}{{\sum\limits_{i = 1}^{n_{1}}{w_{i}^{(1)}{c_{j}\left( X_{({i,j})} \right)}}} = {\ldots = {\sum\limits_{i = 1}^{n_{K}}{w_{i}^{(K)}{c_{j}\left( X_{({i,j})} \right)}}}}}{{{\forall j} = 1},\ldots \mspace{14mu},J}} & {{Equation}\mspace{14mu} 20}\end{matrix}$

In the illustrated example of Equation 20, the first constraintindicates maximizing the total entropy across all probabilitydistributions among the K groups; the second constraint indicates thatthe weights for each group must sum to 100%; and the third constraintindicates the weighted jth covariate across all members for each group(i=1, . . . , n_(k)) must all be equal to each other. The functionc_(j)(X) is up to an analyst and could be a first moment (c_(j)(X)=X), asecond moment (c_(j)(X)=X²), or other functions. In some examples, thefunctions to balance may be determined by a user. For example, balanceweighted averages (as disclosed herein), or use “other functions” suchas alternatively or in addition to balancing first-moments to alsobalancing second-moments; or doing apples-to-apples balancing for eachcategory for categorical variables (i.e., you can't have weightedaverage of a categorical variable). In some examples, the weightingengine 110 balances real-valued numbers matching the weighted averages(as disclosed herein), and also balances their variance (matching bothfirst and second moments). However, if categorical variables (apples,oranges, etc.) which have no numerical values are present, the weightingengine 110 utilizes “other functions,” which in this example matchesapples to apples and oranges to oranges. As such, the weightedprobabilities for each individual category match between treatment groupdataset and control group dataset.

In some examples, the formula for w_(i) ^((k)) has a closed formexpression consistent with Equation 21.

$\begin{matrix}{w_{i}^{(k)} = {\exp \left( {\delta_{0}^{(k)} + {\sum\limits_{j = 1}^{J}{\delta_{j}^{(k)}{c_{j}\left( X_{({i,j})} \right)}}}} \right)}} & {{Equation}\mspace{14mu} 21}\end{matrix}$

Where δ_(j) ^((k)) is expressed in a cyclic manner from the originalLagrange Multipliers (λ_(j) ^((k))), as discussed above in connectionwith Equations 10-19, in a manner consistent with Equation 22.

$\begin{matrix}{\delta_{j}^{(k)} = \left\{ \begin{matrix}{\lambda_{0}^{(k)}\ } & {j = 0} \\{\lambda_{j}^{(k)} - \lambda_{j}^{(K)}} & {{j \neq 0},{k = 1}} \\{\lambda_{j}^{(k)} - \lambda_{j}^{({k - 1})}} & {{j \neq 0},{k \neq 1}}\end{matrix} \right.} & {{Equation}\mspace{14mu} 22}\end{matrix}$

For example, let there be J=2 covariates to match among K=3 groups. Theindex of j=0 is always reserved for the normalization constraint, asshown below.

j k 0 1 2 1 λ⁽¹⁾ ₀ ( λ⁽¹⁾ ₁ − λ⁽³⁾ ₁) ( λ⁽¹⁾ ₂ − λ⁽³⁾ ₂) 2 λ⁽²⁾ ₀ ( λ⁽²⁾₁ − λ⁽¹⁾ ₁) ( λ⁽²⁾ ₂ − λ⁽¹⁾ ₂) 3 λ⁽³⁾ ₀ ( λ⁽³⁾ ₁ − λ⁽²⁾ ₁) ( λ⁽²⁾ ₂ −λ⁽¹⁾ ₂)

The values for any particular row (1, 2, 3) are the same multipliersused in (Equation 21). Also, for any column (0, 1, 2) the sum isalgebraically zero as rearranging terms have each multiplier subtractedfrom itself. As such, the weighting engine 110 represents this in amanner consistent with Equation 23.

$\begin{matrix}{{{\sum\limits_{k = 1}^{K}\delta_{j}^{(k)}} = 0}{{{\forall j} = 1},\ldots \mspace{14mu},J}} & {{Equation}\mspace{14mu} 23}\end{matrix}$

This means without loss of generality the weighting engine 110 can setλ_(j) ^((K))=0 for all j. This reduces the number of LagrangeMultipliers in (Equation 22) from K(J+1) to K(J+1)−J. The fundamentalreason for this is that for any particular covariate where a=b and b=c,the weighting engine 110 does not have to specify that c should equal a,it is automatically satisfied by the transitive relation of equality.This reduces the number of constraints, and therefore LagrangeMultipliers, by one, which in return improves the operational efficiencyof the operating system. As this is true for each covariate, a resultingreduction of J constraints are satisfied.

In some examples, if the same covariates are used within a propensityscore as is with the weighting described above in connection withEquations 5-19, the weighted multinomial propensity score would equal1/K for each individual (i).

In a similar manner, if the ith individual in the kth group has apotential outcome of Y_(i) ^((k)) the weighting engine 110 determinescausal differences ({circumflex over (γ)}) between two different groups(k1, k2) in a manner consistent with Equation 24.

$\begin{matrix}{{\hat{\gamma}}^{({\lbrack{k_{1} - k_{2}}\rbrack})} = {\left( {\sum\limits_{i = 1}^{n_{k_{1}}}{w_{i}^{(k_{1})}Y_{i}^{(k_{1})}}} \right) - \left( {\sum\limits_{i = 1}^{n_{k_{2}}}{w_{i}^{(k_{2})}Y_{i}^{(k_{2})}}} \right)}} & {{Equation}\mspace{14mu} 24}\end{matrix}$

In the illustrated examples of (Equation 21) and (Equation 23), theweighting engine 110 solves for δ_(j) ^((k)) directly, without anyreference to the original λ_(j) ^((k)) which the δ defined. Both the λand the δ act as Lagrange Multipliers. The difference is that theformulation using the δ automatically takes into account the circularequalities (A-B=0, B−C=0, etc.). This is illustrated in (Equation 23)stating that for each covariate the sum of the δ for the covariate mustbe zero across all groups. As such, one of them can be defined as thenegative sum of the other values.

As such, the weighting engine 110 is capable of filtering out redundantvariables which are known functions of other variables. In someexamples, if K groups are to be balanced with J covariates, there are(K−1)J unknown variables which must be solved, independent of the numberof individuals in each group.

In the illustrated example of Equation 25, the weighting engine 110removes the δ₀ factor from Equation 21, and arbitrarily assigned lastgroup k=K to have their values equal the negative sum of the others toensure that (Equation 23) is satisfied.

$\begin{matrix}{{\overset{\sim}{w}}_{i}^{(k)} = \left\{ \begin{matrix}{\exp \left( {\sum\limits_{j = 1}^{J}{\delta_{j}^{(k)}{c_{j}\left( X_{({i,j})} \right)}}} \right)} & {{k = L},\ldots \mspace{14mu},{K - 1}} \\{\exp \left( {\sum\limits_{j = 1}^{J}{\left( {- {\sum\limits_{m = 1}^{K - 1}\delta_{j}^{(m)}}} \right){c_{j}\left( X_{({i,j})} \right)}}} \right)} & {k = K}\end{matrix} \right.} & {{Equation}\mspace{14mu} 25}\end{matrix}$

In the illustrated example of Equation 25 the (˜) represents theun-normalized state. For example, an un-normalized weight ({tilde over(w)}_(i)) equaling [1 2 1] which has a sum=4 would result in anormalized weight (w) equaling [0.25 0.5 0.25], after the normalizationprocesses described in more detail below. The weighting engine 110normalizes each weight, depending on how many weights are in each groupin a manner consistent with Equation 26.

$\begin{matrix}{w_{i}^{(k)} = \frac{{\overset{\sim}{w}}_{i}^{(k)}}{\sum\limits_{i = 1}^{n_{k}}{\overset{\sim}{w}}_{i}^{(k)}}} & {{Equation}\mspace{14mu} 26}\end{matrix}$

The example weighting engine 110 determines an equality structure bysolving the equations represented in Equation 27.

$\begin{matrix}{{{{\sum\limits_{i = 1}^{n_{1}}{w_{i}^{(1)}{c_{j}\left( X_{({i,j})} \right)}}} = {\sum\limits_{i = 1}^{n_{2}}{w_{i}^{(2)}{c_{j}\left( X_{({i,j})} \right)}}}}{{\sum\limits_{i = 1}^{n_{2}}{w_{i}^{(2)}{c_{j}\left( X_{({i,j})} \right)}}} = {\sum\limits_{i = 1}^{n_{3}}{w_{i}^{(3)}{c_{j}\left( X_{({i,j})} \right)}}}}}\vdots {{\sum\limits_{i = 1}^{n_{({K - 1})}}{w_{i}^{(1)}{c_{j}\left( X_{({i,j})} \right)}}} = {\sum\limits_{i = 1}^{n_{K}}{w_{i}^{(K)}{c_{j}\left( X_{({i,j})} \right)}}}}} & {{Equation}\mspace{14mu} 27}\end{matrix}$

The equations in Equation 27 represents that for covariate j all Ktreatment group datasets must have the same weighted average (A=B, B=C,C=D, etc). Additionally, the same equality structure must hold for allother covariates.

For example, the weighting engine 110 may receive the following threesets of data from the covariate engine 108 representing three differentgroups of individuals exposed to three different advertisements.

Third Group index Covariates Outcome (Y³) 1 [10,9] 9 2  [1,1] 5 3  [8,4]10 4  [9,3] 2

First Group Outcome index Covariates (Y²) 1 [2,3] 2 2 [8,5] 4 3 [4,1] 1

First Group index Covariates Outcome (Y¹) 1 [2,3] 2 2 [8.5] 4 3 [4,1] 1

In the above datasets, there are K=3 groups with J=2 covariates tobalance. As such, the weighting engine 110 must have (K−1)J=4 unknownvariables to solve for, which as a result of (Equation 25) are δ₁ ⁽¹⁾δ₂⁽¹⁾δ₁ ⁽²⁾δ₂ ⁽²⁾. The weighting engine 110 solves for Equation 25 in amanner consistent with Equation 28.

$\begin{matrix}{{{\overset{\sim}{w}}_{1}^{(1)} = {\exp \left( {{\delta_{1}^{(1)}(2)} + {\delta_{2}^{(1)}(3)}} \right)}}{{\overset{\sim}{w}}_{2}^{(1)} = {\exp \left( {{\delta_{1}^{(1)}(8)} + {\delta_{2}^{(1)}(5)}} \right)}}{{\overset{\sim}{w}}_{3}^{(1)} = {\exp \left( {{\delta_{1}^{(1)}(4)} + {\delta_{2}^{(1)}(1)}} \right)}}\overset{\_}{{\overset{\sim}{w}}_{1}^{(2)} = {\exp \left( {{\delta_{1}^{(2)}(4)} + {\delta_{2}^{(2)}(8)}} \right)}}\underset{\_}{{\overset{\sim}{w}}_{2}^{(2)} = {\exp \left( {{\delta_{1}^{(2)}(7)} + {\delta_{2}^{(2)}(1)}} \right)}}{{\overset{\sim}{w}}_{1}^{(3)} = {\exp \left( {{{- \left( {\delta_{1}^{(1)} + \delta_{1}^{(2)}} \right)}\left( {10} \right)} + {{- \left( {\delta_{2}^{(1)} + \delta_{2}^{(2)}} \right)}(9)}} \right)}}{{\overset{\sim}{w}}_{2}^{(3)} = {\exp \left( {{{- \left( {\delta_{1}^{(1)} + \delta_{1}^{(2)}} \right)}(1)} + {{- \left( {\delta_{2}^{(1)} + \delta_{2}^{(2)}} \right)}(1)}} \right)}}{{\overset{\sim}{w}}_{3}^{(3)} = {\exp \left( {{{- \left( {\delta_{1}^{(1)} + \delta_{1}^{(2)}} \right)}(8)} \pm {\left( {\delta_{2}^{(1)} + \delta_{2}^{(2)}} \right)(4)}} \right)}}{{\overset{\sim}{w}}_{4}^{(3)} = {\exp \left( {{{- \left( {\delta_{1}^{(1)} + \delta_{1}^{(2)}} \right)}(9)} + {{- \left( {\delta_{2}^{(1)} + \delta_{2}^{(2)}} \right)}(3)}} \right)}}} & {{Equation}\mspace{14mu} 28}\end{matrix}$

The last of the expressions is stating that the unknown coefficient isthe negative of the sum of the other coefficients for that covariate.This is identical to the (Equation 23) constraint.

The weighting engine 110 calculates the normalized weights in a mannerconsistent with Equation 29, subject to the constraints of Equation 30.

$\begin{matrix}{{w_{1}^{(1)} = {{\overset{\sim}{w}}_{1}^{(1)}/\left( {{\overset{\sim}{w}}_{1}^{(1)} + {\overset{\sim}{w}}_{2}^{(1)} + {\overset{\sim}{w}}_{3}^{(1)}} \right)}}{w_{2}^{(1)} = {{\overset{\sim}{w}}_{2}^{(1)}/\left( {{\overset{\sim}{w}}_{1}^{(1)} + {\overset{\sim}{w}}_{2}^{(1)} + {\overset{\sim}{w}}_{3}^{(1)}} \right)}}\underset{\_}{w_{3}^{(1)} = {{\overset{\sim}{w}}_{3}^{(1)}/\left( {{\overset{\sim}{w}}_{1}^{(1)} + {\overset{\sim}{w}}_{2}^{(1)} + {\overset{\sim}{w}}_{3}^{(1)}} \right)}}{w_{1}^{(2)} = {{\overset{\sim}{w}}_{1}^{(2)}/\left( {{\overset{\sim}{w}}_{1}^{(2)} + {\overset{\sim}{w}}_{2}^{(2)}} \right)}}{w_{2}^{(2)} = {{\overset{\sim}{w}}_{2}^{(2)}/\left( {{\overset{\sim}{w}}_{1}^{(2)} + {\overset{\sim}{w}}_{2}^{(2)}} \right)}}\overset{\_}{w_{1}^{(3)} = {{\overset{\sim}{w}}_{1}^{(3)}/\left( {{\overset{\sim}{w}}_{1}^{(3)} + {\overset{\sim}{w}}_{2}^{(3)} + {\overset{\sim}{w}}_{3}^{(3)} + {\overset{\sim}{w}}_{4}^{(3)}} \right)}}{w_{2}^{(3)} = {{\overset{\sim}{w}}_{2}^{(3)}/\left( {{\overset{\sim}{w}}_{1}^{(3)} + {\overset{\sim}{w}}_{2}^{(3)} + {\overset{\sim}{w}}_{3}^{(3)} + {\overset{\sim}{w}}_{4}^{(3)}} \right)}}{w_{3}^{(3)} = {{\overset{\sim}{w}}_{3}^{(3)}/\left( {{\overset{\sim}{w}}_{1}^{(3)} + {\overset{\sim}{w}}_{2}^{(3)} + {\overset{\sim}{w}}_{3}^{(3)} + {\overset{\sim}{w}}_{4}^{(3)}} \right)}}{w_{4}^{(3)} = {{\overset{\sim}{w}}_{4}^{(3)}/\left( {{\overset{\sim}{w}}_{1}^{(3)} + {\overset{\sim}{w}}_{2}^{(3)} + {\overset{\sim}{w}}_{3}^{(3)} + {\overset{\sim}{w}}_{4}^{(3)}} \right)}}} & {{Equation}\mspace{14mu} 29}\end{matrix}$

The constraints that must be satisfied are illustrated in Equation 30.

(w ₁ ⁽¹⁾)(2)+(w ₂ ⁽¹⁾)(8)+(w ₃ ⁽¹⁾)(4)=(w ₁ ⁽²⁾)(4)+(w ₂ ⁽²⁾)(7)

(w ₁ ⁽²⁾)(4)+(w ₂ ⁽²⁾)(7)=(w ₁ ⁽³⁾)(10)+(w ₂ ⁽³⁾)(1)+(w ₃ ⁽³⁾)(8)+(w ₄⁽³⁾)(9)

(w ₁ ⁽¹⁾)(3)+(w ₂ ⁽¹⁾)(5)+(w ₃ ⁽¹⁾)(1)=(w ₁ ⁽²⁾)(8)+(w ₂ ⁽²⁾)(1)

(w ₁ ⁽²⁾)(8)+(w ₂ ⁽²⁾)(1)=(w ₁ ⁽³⁾)(9)+(w ₂ ⁽³⁾)(1)+(w ₃ ⁽³⁾)(4)+(w ₄⁽³⁾)(3)  Equation 30:

In the illustrated example of Equation 30, each equation represents thefollowing: 1) Weighted average of the first covariate in group 1 mustequal the weighted average of the first covariate in group 2; 2)Weighted average of the first covariate in group 2 must equal theweighted average of the first covariate in group 3; 3) Weighted averageof the second covariate in group 1 must equal the weighted average ofthe second covariate in group 2; 4) Weighted average of the secondcovariate in group 2 must equal the weighted average of the secondcovariate in group 3

As such, there are four equations in four unknowns δ₁ ⁽¹⁾δ₂ ⁽¹⁾δ₁ ⁽²⁾δ₂⁽²⁾, which the weighting engine 110 solves for utilizing any non-linearmultivariate solver, resulting in Equation 31.

δ₁ ⁽¹⁾=+0.152074

δ₂ ⁽¹⁾=+0.0656848

δ₁ ⁽²⁾=−0.0548394

δ₂ ⁽²⁾=−0.0937144  Equation 31:

The weighting engine 110 inputs the results from Equation 30 into(Equation 28) to determine the un-normalized weights in a mannerconsistent with Equation 32.

$\begin{matrix}{{{\overset{\sim}{w}}_{1}^{(1)} = 1.65071}{{\overset{\sim}{w}}_{2}^{(1)} = {{4.6}8807}}{{\overset{\sim}{w}}_{3}^{(1)} = {{1.9}6204}}\overset{\_}{{\overset{\sim}{w}}_{1}^{(2)} = {{0.3}79435}}\underset{\_}{{\overset{\sim}{w}}_{2}^{(2)} = 0.620276}{{\overset{\sim}{w}}_{1}^{(3)} = {0.486711}}{{\overset{\sim}{w}}_{2}^{(3)} = {{0.9}33135}}{{\overset{\sim}{w}}_{3}^{(3)} = {{0.5}13881}}{{\overset{\sim}{w}}_{4}^{(3)} = {{0.4}53379}}} & {{Equation}\mspace{14mu} 32}\end{matrix}$

The weighting engine 110 determines the normalized set of weights byinputting the results of Equation 32 into Equation 26 to produce resultsin a manner consistent with Equation 33.

$\begin{matrix}{{w_{1}^{(1)} = {{0.1}98861}}{w_{2}^{(1)} = {{0.5}64772}}{w_{2}^{(1)} = {{0.2}36367}}\overset{\_}{w_{1}^{(2)} = {{0.3}79435}}\frac{w_{2}^{(2)} = {{0.6}20276}}{w_{1}^{(3)} = {{0.2}03892}}{w_{2}^{(3)} = {{0.3}90906}}{w_{3}^{(3)} = {{0.2}15274}}{w_{4}^{(3)} = {{0.1}89928}}} & {{Equation}\mspace{14mu} 33}\end{matrix}$

The weighting response engine 112 determines the resulting causal effect(e.g., estimated average outcomes (Y _(W))) in a manner consistent withEquation 34 (e.g., in a manner consistent with Equations 18 and 19).

Y _(W) ⁽¹⁾=2.89318

Y _(W) ⁽²⁾=5.75909

Y _(W) ⁽³⁾=6.32215  Equation 34:

The resulting estimated average outcomes are the estimated averageoutcomes for each treatment (e.g., advertisement campaign) as if theobserved individuals (e.g., individuals in the treatment group dataset)actually were in a randomized experiment. As such, the weightingresponse engine 112 determines the net positive or negative differenceacross different treatments. For example, if the estimated averageoutcomes were dollar purchases and treatment 1 (e.g., advertisement 1)corresponds to “control” and treatment 2 (e.g., advertisement 2)correspond to television advertising, the weighting response engine 112would calculate a $2.86 (5.75−2.89) increase in purchases by anindividual viewing the television advertisement as opposed to notviewing the advertisement at all (e.g., control). In some examples,treatment 3 (e.g., advertisement 3) could be an internet advertisement.As such, combinations of treatments can also be estimated by theweighting response engine 112. For example, if treatment (1)=None,treatment (2)=TV only advertisement, treatment (3)=Internet onlyadvertisement, and treatment (4)=Both TV and Internet advertisement, theweighting response engine 112 estimates the causal effect of seeingadvertisements from one or another, and/or a combined effect of seeingan advertisement on multiple devices. This is beneficial because itallows advertisers to determine which combination among multipleplatforms produce the most impact, and allows advertisers to adjusttheir advertising strategies accordingly taking into account budgetaryand business decisions.

In some examples, when the treatment/control data stores(s) 104 includedesign weights (d) (e.g., sampling weights), the analysis engine 102substitutes the following equations into the processes disclosed abovein connection with Equations 1-34. For example, the analysis engine 102substitutes Equation 1a for Equation 1, Equation 3a for Equation 3,Equation 5a for Equation 5, Equation 7a for Equation 7, Equation 8a forEquation 8, Equation 13a for Equation 13, Equation 20a for Equation 20,Equation 21a for Equation 21, and Equation 25a for Equation 25. As such,the analysis engine 102 determines causal effects for treatment groupdatasets and control group datasets that include design weights (d) in amanner consistent with Equations 1a, 3a, 5a, 7a, 8a, 13a, 20a, 21a, and25a while maintaining the efficiencies and operation of the analysisengine 102 disclosed above in connection with Equations 1-34. That is,Equations 1a, 3a, 5a, 7a, 8a, 13a, 20a, 21a, and 25a are substitutedinto the above disclosure of Equations 1-34 to accommodate the designweights (d) of the treatment group datasets and the control groupdatasets.

$\begin{matrix}{\mspace{20mu} {H = {- {\sum\limits_{Z_{i} = 0}{w_{i}{\log \left( \frac{w_{i}}{d_{i}} \right)}}}}}} & {{Equation}\mspace{14mu} 1a} \\{\mspace{20mu} {H = {- {\sum\limits_{Z_{i} = 1}{v_{i}{\log \left( \frac{v_{i}}{d_{i}} \right)}}}}}} & {{Equation}\mspace{14mu} 3a} \\{\mspace{20mu} {H = {\left( {- {\sum\limits_{Z_{i} = 1}{v_{i}{\log \left( \frac{v_{i}}{d_{i}} \right)}}}} \right) + \left( {- {\sum\limits_{Z_{i} = 0}{w_{i}{\log \left( \frac{w_{i}}{d_{i}} \right)}}}} \right)}}} & {{Equation}\mspace{14mu} 5a} \\{\mspace{20mu} {H = {- {\sum\limits_{i = 1}^{n_{1} + n_{0}}{p_{i}{\log \left( \frac{p_{i}}{d_{i}} \right)}}}}}} & {{Equation}\mspace{14mu} 7a} \\{\mspace{20mu} {{v_{i} = {d_{i}{\exp\left( {\lambda_{0}^{(1)} + {\sum\limits_{j = 1}^{m}{\lambda_{j}{c_{j}\left( x_{i} \right)}}}} \right)}}}\mspace{20mu} {w_{i} = {d_{i}{\exp\left( {\lambda_{0}^{(0)} + {\sum\limits_{j = 1}^{m}{\left( {- \lambda_{j}} \right){c_{j}\left( x_{i} \right)}}}} \right)}}}}} & {{Equation}\mspace{14mu} 8a} \\{\mspace{20mu} {H = {- {\sum\limits_{i = 1}^{n_{2} + n_{1} + n_{0}}{p_{i}{\log \left( \frac{p_{i}}{d_{i}} \right)}}}}}} & {{Equation}\mspace{14mu} 13a} \\{\mspace{20mu} {{KL} = {\sum\limits_{k = 1}^{K}\left( {\sum\limits_{i = 1}^{n_{k}}{w_{i}^{(k)}{\log \left( \frac{w_{i}^{(k)}}{d_{i}^{(k)}} \right)}}} \right)}}} & {{Equation}\mspace{14mu} 20a} \\{\mspace{20mu} {w_{i}^{(k)} = {d_{i}^{(k)}{\exp\left( {\delta_{0}^{(k)} + {\sum\limits_{j = 1}^{J}{\delta_{j}^{(k)}{c_{j}\left( X_{({i,j})} \right)}}}} \right)}}}} & {{Equation}\mspace{14mu} 21a} \\{{\overset{\sim}{w}}_{i}^{(k)} = \left\{ \begin{matrix}{d_{i}^{(k)}{\exp \left( {\sum\limits_{j = 1}^{J}{\delta_{j}^{(k)}{c_{j}\left( X_{({i,j})} \right)}}} \right)}} & {{k = 1},\ldots \mspace{14mu},{K - 1}} \\{d_{i}^{(k)}{\exp \left( {\sum\limits_{j = 1}^{J}{\left( {- {\sum\limits_{k = 1}^{K - 1}\delta_{j}^{(m)}}} \right){c_{j}\left( X_{({i,j})} \right)}}} \right)}} & {k = K}\end{matrix} \right.} & {{Equation}\mspace{14mu} 25a}\end{matrix}$

While an example manner of implementing example environment inaccordance with the teachings of this disclosure is illustrated in FIG.1, one or more of the elements, processes and/or devices illustrated inFIG. 1 may be combined, divided, re-arranged, omitted, eliminated and/orimplemented in any other way. Further, the example treatment/controldata interface 106, the example covariate engine 108, the exampleweighting engine 110, the example weighting response engine 112, theexample report generator 114, and/or, more generally, the exampleanalysis engine 102 of FIG. 1 may be implemented by hardware, software,firmware and/or any combination of hardware, software and/or firmware.Thus, for example, any of the example treatment/control data interface106, the example covariate engine 108, the example weighting engine 110,the example weighting response engine 112, the example report generator114, and/or, more generally, the example analysis engine 102 of FIG. 1could be implemented by one or more analog or digital circuit(s), logiccircuits, programmable processor(s), application specific integratedcircuit(s) (ASIC(s)), programmable logic device(s) (PLD(s)) and/or fieldprogrammable logic device(s) (FPLD(s)). When reading any of theapparatus or system claims of this patent to cover a purely softwareand/or firmware implementation, at least one of the exampletreatment/control data interface 106, the example covariate engine 108,the example weighting engine 110, the example weighting response engine112, the example report generator 114, and/or, more generally, theexample analysis engine 102 of FIG. 1 is/are hereby expressly defined toinclude a non-transitory computer readable storage device or storagedisk such as a memory, a digital versatile disk (DVD), a compact disk(CD), a Blu-ray disk, etc. including the software and/or firmware.Further still, the example environment 100 and/or the example analysisengine 102 may include one or more elements, processes and/or devices inaddition to, or instead of, those illustrated in FIG. 1, and/or mayinclude more than one of any or all of the illustrated elements,processes and devices.

Flowcharts representative of example machine readable instructions forimplementing the example treatment/control data interface 106, theexample covariate engine 108, the example weighting engine 110, theexample weighting response engine 112, the example report generator 114,and/or, more generally, the example analysis engine 102 of FIG. 1 areshown in FIGS. 2-5. In these examples, the machine readable instructionscomprise one or more programs for execution by a processor, such as theprocessor 612 shown in the example processor platform 600 discussedbelow in connection with FIG. 6. The one or more programs, or portion(s)thereof, may be embodied in software stored on a non-transitory computerreadable storage medium such as a CD-ROM, a floppy disk, a hard drive, adigital versatile disk (DVD), a Blu-ray Disk™, or a memory associatedwith the processor 612, but the entire program or programs and/or partsthereof could alternatively be executed by a device other than theprocessor 612 and/or embodied in firmware or dedicated hardware (e.g.,implemented by an ASIC, a PLD, an FPLD, discrete logic, etc.). Further,although the example program(s) is(are) described with reference to theflowcharts illustrated in FIGS. 2-5, many other methods of implementingthe example treatment/control data interface 106, the example covariateengine 108, the example weighting engine 110, the example weightingresponse engine 112, the example report generator 114, and/or, moregenerally, the example analysis engine 102 of FIG. 1 may alternativelybe used. For example, with reference to the flowcharts illustrated inFIGS. 2-5, the order of execution of the blocks may be changed, and/orsome of the blocks described may be changed, eliminated, combined and/orsubdivided into multiple blocks. Additionally or alternatively, any orall of the blocks may be implemented by one or more hardware circuits(e.g., discrete and/or integrated analog and/or digital circuitry, aField Programmable Gate Array (FPGA), an Application Specific Integratedcircuit (ASIC), a comparator, an operational-amplifier (op-amp), a logiccircuit, etc.) structured to perform the corresponding operation withoutexecuting software or firmware.

As mentioned above, the example processes of FIGS. 2-5 may beimplemented using coded instructions (e.g., computer and/or machinereadable instructions) stored on a non-transitory computer and/ormachine readable medium such as a hard disk drive, a flash memory, aread-only memory, a compact disk, a digital versatile disk, a cache, arandom-access memory and/or any other storage device or storage disk inwhich information is stored for any duration (e.g., for extended timeperiods, permanently, for brief instances, for temporarily buffering,and/or for caching of the information). As used herein, the termnon-transitory computer readable medium is expressly defined to includeany type of computer readable storage device and/or storage disk and toexclude propagating signals and to exclude transmission media.“Including” and “comprising” (and all forms and tenses thereof) are usedherein to be open ended terms. Thus, whenever a claim lists anythingfollowing any form of “include” or “comprise” (e.g., comprises,includes, comprising, including, etc.), it is to be understood thatadditional elements, terms, etc. may be present without falling outsidethe scope of the corresponding claim. As used herein, when the phrase“at least” is used as the transition term in a preamble of a claim, itis open-ended in the same manner as the term “comprising” and“including” are open ended. Also, as used herein, the terms “computerreadable” and “machine readable” are considered equivalent unlessindicated otherwise.

An example program 200 that may be executed in the example environment100 of FIG. 1 to leverage the analysis engine 102 to determine causaleffects for audience measurement is illustrated in FIG. 2. In theillustrated example of FIG. 2, the example covariate engine 108determines a first set of covariates to be processed for a treatmentgroup dataset of an advertisement campaign and a second set ofcovariates to be processed for a control group dataset of theadvertisement campaign, the treatment group dataset is indicative ofbeing exposed to an advertisement, while the control group dataset isindicative of not being exposed to an advertisement (block 202). Forexample, the covariate engine 108 may determine a covariate (e.g., age,income, etc.) to balance that is available across both the treatmentgroup dataset and the control group dataset. The example weightingengine 110 may calculate a first set of weights for the first set ofcovariates corresponding to the treatment group dataset and a second setof weights for the second set of covariates corresponding to the controlgroup dataset using maximum entropy, the first set of weights is toequal the second set of weights (block 204). For example, the weightingengine 110 may calculate the first set of weights and the second set ofweights in a manner consistent with example Equations 5 and 6. Theexample weighting response engine 112 calculates a weighted response forthe treatment group dataset and a weighted response for the controlgroup dataset based on the first set of weights and the second set ofweights (block 206). For example, the weighting response engine 112 maybypass multivariate reweighting by calculating the weighted response forthe treatment group dataset and the weighted response for the controlgroup dataset in a manner consistent with example Equations 7-10. Theexample weighting response engine 112 determines an average treatmenteffect based on a difference between the weighted response for thetreatment dataset and the weighted response for the control dataset(block 208). For example, the weighting response engine 112 maydetermine the average treatment effect in manner consistent with exampleEquation 12. In some example, when no more treatment and control groupdatasets have been identified for processing, the example program 200ends.

FIG. 3 illustrates an example of performing the processes of block 206to leverage the analysis engine 102 to determine causal effects foraudience measurement. In the illustrated example of FIG. 3, the exampleweighting engine 110 maps the first set of weights for the first set ofcovariates to a uniform weighting identifier based on selecting a firstLagrange multiplier to represent the first set of weights, and maps thesecond set of weights for the second set of covariates to the uniformweighting identifier based on selecting a second Lagrange multiplier torepresent the second set of weights (block 302). For example, theweighting engine 110 maps the first and second set of weights to theuniform weighting identifier in manner consistent with example Equation7. The example weighting engine 110 determines a constraint matrix basedon the first set of covariates, the second set of covariates, and theuniform weighting identifier (block 304). For example, the weightingengine 110 may determine the constraint matrix in a manner consistentwith example Equation 9. The weighting engine 110 solves for the firstand second Lagrange multipliers by applying the constraint matrix to amaximum entropy algorithm utilizing the uniform weighting identifier(block 306). For example, the weighting engine 110 may solve theLagrange multipliers in a manner consistent with example Equation 10.The example weighting engine 110 calculates a weighted response for thetreatment group dataset and a weighted response for the control groupdataset based on the Lagrange multipliers (block 308). For example, theweighting engine 110 calculates the weighted responses in a mannerconsistent with example Equation 10. The process returns to FIG. 2

Unlike the illustrated example program 200 of FIG. 2, the illustratedexample of FIG. 4 considers when a single treatment group dataset and asingle control group dataset are present. An example program 400 thatmay be executed in the example environment 100 of FIG. 1 to leverage theanalysis engine 102 to determine causal effects for audience measurementis illustrated in FIG. 4. In the illustrated example of FIG. 4, theprogram 400 includes instructions for execution by the analysis engine102 when a single treatment group dataset and a single control groupdataset are present. In the illustrated example of FIG. 4, the analysisengine 102 obtains treatment group dataset and control group dataset,the treatment group dataset and control group dataset include covariatesof each individual (age, gender, etc.) and the individuals response totreatment (e.g., buying of a product from exposure to advertisement)(block 402). For example, the treatment/control data interface mayobtain treatment and control group datasets from the treatment/controldata store(s) 104. The covariate engine 108 determines which covariatesto balance (block 404). For example, the covariate engine 108 maydetermine a covariate (e.g., age, income, etc.) to balance that isavailable across both the treatment group dataset and the control groupdataset. The example weighting engine 110 uses the Maximum Entropyformulation to find weights for each individual in the treatment andcontrol group datasets so the weighted covariates equal between bothgroups (block 406). For example, the weighting engine 110 may calculatethe first stet of weights and the second set of weights in a mannerconsistent with example Equations 5 and 6. The example weightingresponse engine 112 calculates a weighted response for the treatment andcontrol group datasets based on the weights from the Maximum Entropyformulation (block 408). For example, the weighting response engine 112may calculate the weighted response for the treatment group dataset andthe weighted response for the control group dataset in a mannerconsistent with example Equations 7-10. The example weighting responseengine 112 determines the causal effect of the treatment based on adifference between the weighted response for the treatment group datasetand the weighted response for the control group dataset (block 410). Forexample, the weighting response engine 112 may determine the causaleffect of the treatment in a manner consistent with example Equation 12.In some example, when no more treatment and control group datasets havebeen identified for processing, the example program 400 ends.

An example program 500 that may be executed in the example environment100 of FIG. 1 to leverage the analysis engine 102 to determine causaleffects for audience measurement is illustrated in FIG. 5. In theillustrated example of FIG. 5, the program 500 includes instructions forexecution by the analysis engine 102 when multiple treatment groupdatasets and/or multiple control group datasets are present. In theillustrated example of FIG. 5, the analysis engine 102 obtains alltreatment group datasets and control group datasets, the treatment groupdatasets and control group datasets including covariates of eachindividual (age, gender, etc.) and the individuals response to treatment(e.g., buying of a product from exposure to advertisement) (block 502).For example, the treatment/control data interface may obtain treatmentand control group datasets from the treatment/control data store(s) 104.The covariate engine 108 determines which covariates to balance (block504). For example, the covariate engine 108 may determine a covariate(e.g., age, income, etc.) to balance that is available across all thetreatment group datasets and the control group datasets. The exampleweighting engine 110 uses the Maximum Entropy formulation to findweights for each individual in the treatment and control group datasetsso the weighted covariates equal across all groups (block 506). Forexample, the weighting engine 110 may calculate the weights in a mannerconsistent with example Equation 13. The example weighting responseengine 112 calculates a weighted response for the treatment and controlgroup datasets based on the weights from the Maximum Entropy formulation(block 508). For example, the weighting response engine 112 maycalculate the weighted response for the treatment group dataset and theweighted response for the control group dataset in a manner consistentwith example Equations 13-16. The example weighting response engine 112determines the causal effect between each treatment group dataset, orbetween a treatment group dataset and a control group dataset based on adifference between the weighted response for a specific group datasetand the weighted response for a different group dataset (block 510). Forexample, the weighting response engine 112 may determine the causaleffect of the treatment in a manner consistent with example Equations 18and 19. In some example, when no more treatment and control groupdatasets have been identified for processing, the example program 500ends.

FIG. 6 is a block diagram of an example processor platform 600structured to execute the example instructions of FIGS. 2-5 to implementthe example analysis engine 102. The processor platform 600 can be, forexample, a server, a personal computer, a mobile device (e.g., a cellphone, a smart phone, a tablet such as an iPad™), a personal digitalassistant (PDA), an Internet appliance, a DVD player, a CD player, adigital video recorder, a Blu-ray player, a gaming console, a personalvideo recorder, a set top box a digital camera, or any other type ofcomputing device.

The processor platform 600 of the illustrated example includes aprocessor 612. The processor 612 of the illustrated example is hardware.For example, the processor 612 can be implemented by one or moreintegrated circuits, logic circuits, microprocessors or controllers fromany desired family or manufacturer. The hardware processor 612 may be asemiconductor based (e.g., silicon based) device. In this example, theprocessor 612 implements the example analysis engine 102.

The processor 612 of the illustrated example includes a local memory 613(e.g., a cache). The processor 612 of the illustrated example is incommunication with a main memory including a volatile memory 614 and anon-volatile memory 616 via a link 618. The link 618 may be implementedby a bus, one or more point-to-point connections, etc., or a combinationthereof. The volatile memory 614 may be implemented by SynchronousDynamic Random Access Memory (SDRAM), Dynamic Random Access Memory(DRAM), RAMBUS Dynamic Random Access Memory (RDRAM) and/or any othertype of random access memory device. The non-volatile memory 616 may beimplemented by flash memory and/or any other desired type of memorydevice. Access to the main memory 614, 616 is controlled by a memorycontroller.

The processor platform 600 of the illustrated example also includes aninterface circuit 620. The interface circuit 620 may be implemented byany type of interface standard, such as an Ethernet interface, auniversal serial bus (USB), and/or a PCI express interface.

In the illustrated example, one or more input devices 622 are connectedto the interface circuit 620. The input device(s) 622 permit(s) a userto enter data and commands into the processor 612. The input device(s)can be implemented by, for example, an audio sensor, a microphone, acamera (still or video), a keyboard, a button, a mouse, a touchscreen, atrack-pad, a trackball, a trackbar (such as an isopoint), a voicerecognition system and/or any other human-machine interface. Also, manysystems, such as the processor platform 600, can allow the user tocontrol the computer system and provide data to the computer usingphysical gestures, such as, but not limited to, hand or body movements,facial expressions, and face recognition.

One or more output devices 624 are also connected to the interfacecircuit 620 of the illustrated example. The output devices 624 can beimplemented, for example, by display devices (e.g., a light emittingdiode (LED), an organic light emitting diode (OLED), a liquid crystaldisplay, a cathode ray tube display (CRT), a touchscreen, a tactileoutput device, a printer and/or speakers). The interface circuit 620 ofthe illustrated example, thus, typically includes a graphics drivercard, a graphics driver chip or a graphics driver processor.

The interface circuit 620 of the illustrated example also includes acommunication device such as a transmitter, a receiver, a transceiver, amodem and/or network interface card to facilitate exchange of data withexternal machines (e.g., computing devices of any kind) via a network626 (e.g., an Ethernet connection, a digital subscriber line (DSL), atelephone line, coaxial cable, a cellular telephone system, etc.).

The processor platform 600 of the illustrated example also includes oneor more mass storage devices 628 for storing software and/or data.Examples of such mass storage devices 628 include floppy disk drives,hard drive disks, compact disk drives, Blu-ray disk drives, RAID(redundant array of independent disks) systems, and digital versatiledisk (DVD) drives.

Coded instructions 632 corresponding to the instructions of FIGS. 2-5may be stored in the mass storage device 628, in the volatile memory614, in the non-volatile memory 616, in the local memory 613 and/or on aremovable tangible computer readable storage medium, such as a CD or DVD636.

From the foregoing, it will be appreciated that example methods,apparatus and articles of manufacture have been disclosed that improvethe operating efficiency of computing devices by calculating weights forcovariates simultaneously. Such disclosed examples increase thecomputational efficiency of determining causal effects for audiencemeasurement by reducing the amount of processing cycles required tore-process another set of covariates for a treatment/control groupdataset. The disclosed methods, apparatus and articles of manufactureare accordingly directed to one or more improvement(s) in thefunctioning of a computer.

The following paragraphs provide various examples of the examplesdisclosed herein.

Example 1 can be a weighting engine to calculate a first set of weightsfor a first set of covariates corresponding to a treatment dataset and asecond set of weights for a second set of covariates corresponding to acontrol dataset using maximum entropy, the first set of weights to equalthe second set of weights; and a weighting response engine to calculatea weighted response for the treatment dataset and a weighted responsefor the control dataset by: mapping the first set of weights and thesecond set of weights to a uniform weighting identifier, determining aconstraint matrix based on the first set of weights, the second set ofweights and the uniform weighting identifier, and bypassing multivariatereweighting by calculating the weighted response for the treatmentdataset and the weighted response for the control dataset by applyingmaximum entropy to the constraint matrix.

Example 2 includes the apparatus of example 1, further including acovariate engine to determine the first set of covariates to beprocessed for the treatment dataset of an advertisement campaign and thesecond set of covariates to be processed for the control dataset of theadvertisement campaign, the treatment dataset indicative of beingexposed to an advertisement, the control dataset indicative of not beingexposed to an advertisement.

Example 3 includes the apparatus of any one of examples 1-2, wherein theweighting engine is to determine the first set of weights independent ofthe second set of weights to increase an operational efficiency of theapparatus.

Example 4 includes the apparatus of any one of examples 1-3, wherein theweighting response engine is to calculate the weighted response for thetreatment dataset and the weighted response for the control dataset sothat the weighted response for the treatment dataset and the weightedresponse for the control dataset are on a common scale.

Example 5 includes the apparatus of any one of examples 1-4, wherein theweighting response engine is to determine an average treatment effect bydetermining a difference between the weighted response for the treatmentdataset and the weighted response for the control dataset, the averagetreatment effect indicative of a potential increase per individualexposed to an advertisement campaign.

Example 6 includes the apparatus of any one of examples 1-5, furtherincluding a report generator to transmit a report to an audiencemeasurement entity.

Example 7 includes the apparatus of any one of examples 1-6, wherein thereport generator is to display the report on a device via a webpage in afirst state with a set of options, the options selectable by a user tochange to the first state.

Example 8 can be a non-transitory computer readable medium comprisinginstructions that, when executed, cause a machine to at least calculatea first set of weights for a first set of covariates corresponding to atreatment dataset and a second set of weights for a second set ofcovariates corresponding to a control dataset using maximum entropy, thefirst set of weights to equal the second set of weights; and calculate aweighted response for the treatment dataset and a weighted response forthe control dataset by: mapping the first set of weights and the secondset of weights to a uniform weighting identifier, determining aconstraint matrix based on the first set of weights, the second set ofweights and the uniform weighting identifier, and bypassing multivariatereweighting by calculating the weighted response for the treatmentdataset and the weighted response for the control dataset by applyingmaximum entropy to the constraint matrix.

Example 9 includes the non-transitory computer readable medium ofexample 8, wherein the instructions further cause the machine todetermine the first set of covariates to be processed for the treatmentdataset of an advertisement campaign and the second set of covariates tobe processed for the control dataset of the advertisement campaign, thetreatment dataset indicative of being exposed to an advertisement, thecontrol dataset indicative of not being exposed to an advertisement.

Example 10 includes the non-transitory computer readable medium of anyone of examples 8-9, wherein the instructions further cause the machineto determine the first set of weights independent of the second set ofweights to increase an operational efficiency of the apparatus.

Example 11 includes the non-transitory computer readable medium of anyone of examples 8-10, wherein the instructions further cause the machineto calculate the weighted response for the treatment dataset and theweighted response for the control dataset so that the weighted responsefor the treatment dataset and the weighted response for the controldataset are on a common scale.

Example 12 includes the non-transitory computer readable medium of anyone of examples 8-11, wherein the instructions further cause the machineto determine an average treatment effect by determining a differencebetween the weighted response for the treatment dataset and the weightedresponse for the control dataset, the average treatment effectindicative of a potential increase per individual exposed to anadvertisement campaign.

Example 13 includes the non-transitory computer readable medium of anyone of examples 8-12, wherein the instructions further cause the machineto transmit a report to an audience measurement entity.

Example 14 includes the non-transitory computer readable medium of anyone of examples 8-13, wherein the instructions further cause the machineto display the report on a device via a webpage in a first state with aset of options, the options selectable by a user to change to the firststate.

Example 15 can be means for calculating a weight to calculate a firstset of weights for a first set of covariates corresponding to atreatment dataset and a second set of weights for a second set ofcovariates corresponding to a control dataset using maximum entropy, thefirst set of weights to equal the second set of weights; and means fordetermining a weighted response to calculate a weighted response for thetreatment dataset and a weighted response for the control dataset by:mapping the first set of weights and the second set of weights to auniform weighting identifier, determining a constraint matrix based onthe first set of weights, the second set of weights and the uniformweighting identifier, and bypassing multivariate reweighting bycalculating the weighted response for the treatment dataset and theweighted response for the control dataset by applying maximum entropy tothe constraint matrix.

Example 16 includes the apparatus of example 15, further including meansfor generating a covariate to determine the first set of covariates tobe processed for the treatment dataset of an advertisement campaign andthe second set of covariates to be processed for the control dataset ofthe advertisement campaign, the treatment dataset indicative of beingexposed to an advertisement, the control dataset indicative of not beingexposed to an advertisement.

Example 17 includes the apparatus of any one of examples 15-16, whereinthe weight calculating means is to determine the first set of weightsindependent of the second set of weights to increase an operationalefficiency of the apparatus.

Example 18 includes the apparatus of any one of examples 15-17, whereinthe weighted response determining means is to calculate the weightedresponse for the treatment dataset and the weighted response for thecontrol dataset so that the weighted response for the treatment datasetand the weighted response for the control dataset are on a common scale.

Example 19 includes the apparatus of any one of examples 15-18, whereinthe weighted response determining means is to determine an averagetreatment effect by determining a difference between the weightedresponse for the treatment dataset and the weighted response for thecontrol dataset, the average treatment effect indicative of a potentialincrease per individual exposed to an advertisement campaign.

Example 20 includes the apparatus of any one of examples 15-19, furtherincluding means for generating a report to display a report on a devicevia a webpage in a first state with a set of options, the optionsselectable by a user to change to the first state.

Example 21 can be calculating, by executing an instruction with aprocessor, a first set of weights for a first set of covariatescorresponding to a treatment dataset and a second set of weights for asecond set of covariates corresponding to a control dataset usingmaximum entropy, the first set of weights to equal the second set ofweights; and calculating, by executing an instruction with theprocessor, a weighted response for the treatment dataset and a weightedresponse for the control dataset by: mapping the first set of weightsand the second set of weights to a uniform weighting identifier,determining a constraint matrix based on the first set of weights, thesecond set of weights and the uniform weighting identifier, andbypassing multivariate reweighting by calculating the weighted responsefor the treatment dataset and the weighted response for the controldataset by applying maximum entropy to the constraint matrix.

Example 22 includes the method of example 21, further includingdetermining the first set of covariates to be processed for thetreatment dataset of an advertisement campaign and the second set ofcovariates to be processed for the control dataset of the advertisementcampaign, the treatment dataset indicative of being exposed to anadvertisement, the control dataset indicative of not being exposed to anadvertisement.

Example 23 includes the method of any one of examples 21-22, furtherincluding determining the first set of weights independent of the secondset of weights to increase an operational efficiency of the apparatus.

Example 24 includes the method of any one of examples 21-23, furtherincluding determining the weighted response for the treatment datasetand the weighted response for the control dataset so that the weightedresponse for the treatment dataset and the weighted response for thecontrol dataset are on a common scale.

Example 25 includes the method of any one of examples 21-24, furtherincluding determining an average treatment effect by determining adifference between the weighted response for the treatment dataset andthe weighted response for the control dataset, the average treatmenteffect indicative of a potential increase per individual exposed to anadvertisement campaign.

Example 26 includes the method of any one of examples 21-25, furtherincluding transmitting a report to an audience measurement entity.

Example 27 includes the method of any one of examples 21-26, furtherincluding displaying the report on a device via a webpage in a firststate with a set of options, the options selectable by a user to changeto the first state.

Although certain example methods, apparatus and articles of manufacturehave been disclosed herein, the scope of coverage of this patent is notlimited thereto. On the contrary, this patent covers all methods,apparatus and articles of manufacture fairly falling within the scope ofthe claims of this patent.

1. An apparatus comprising: a weighting engine to: calculate a first setof weights for a first set of covariates corresponding to a firsttreatment dataset using maximum entropy; calculate a second set ofweights for a second set of covariates corresponding to a secondtreatment dataset using maximum entropy; and calculate a third set ofweights for a third set of covariates corresponding to a control datasetusing maximum entropy, the weighting engine to increase an operationalefficiency of the apparatus by calculating the first set of weights, thesecond set of weights, and the third set of weights independently; aweighting response engine to: calculate a first weighted response forthe first treatment dataset, calculate a second weighted response forthe second treatment dataset, and calculate a third weighted responsefor the control dataset; and determine a causal effect between the firsttreatment dataset and the second treatment dataset based on a differencebetween the first weighted response and the second weighted response;and a report generator to transmit a report to an audience measuremententity, the report generator to cause display of the report on a device,the report including the causal effect, at least one of the weightingengine, the weighting response engine, or the report generatorimplemented by a logic circuit.
 2. The apparatus as defined in claim 1,wherein the first treatment dataset corresponds to a first advertisementcampaign and the second treatment dataset corresponds to a secondadvertisement campaign.
 3. The apparatus as defined in claim 2, whereinthe causal effect is indicative of a potential increase per individualexposed to the first advertisement campaign compared to the secondadvertisement campaign.
 4. (canceled)
 5. The apparatus as defined inclaim 1, wherein the weighting engine is to calculate the first set ofweights, the second set of weights, and the third set of weights in oneprocessor cycle.
 6. The apparatus as defined in claim 1, wherein theweight response engine is to calculate the first weighted response, thesecond weighted response, and the third weighted response by bypassingmultivariate reweighting to improve an operating efficiency of theapparatus by calculating the first weighted response, the secondweighted response, and the third weighted response by applying maximumentropy to a constraint matrix.
 7. (canceled)
 8. A non-transitorycomputer readable medium comprising instructions that, when executed,cause a machine to at least: calculate a first set of weights for afirst set of covariates corresponding to a first treatment dataset usingmaximum entropy; calculate a second set of weights for a second set ofcovariates corresponding to a second treatment dataset using maximumentropy; calculate a third set of weights for a third set of covariatescorresponding to a control dataset using maximum entropy, the first setof weights, the second set of weights, and the third set of weightscalculated independently to increase an operational efficiency of themachine; calculate a first weighted response for the first treatmentdataset, calculate a second weighted response for the second treatmentdataset, and calculate a third weighted response for the controldataset; determine a causal effect between the first treatment datasetand the second treatment dataset based on a difference between the firstweighted response and the second weighted response; and transmit areport to an audience measurement entity to cause display of the reporton a device of the audience measurement entity.
 9. The non-transitorycomputer readable medium as defined in claim 8, wherein the firsttreatment dataset corresponds to a first advertisement campaign and thesecond treatment dataset corresponds to a second advertisement campaign.10. The non-transitory computer readable medium as defined in claim 9,wherein the causal effect is indicative of a potential increase perindividual exposed to the first advertisement campaign compared to thesecond advertisement campaign.
 11. (canceled)
 12. The non-transitorycomputer readable medium as defined in claim 8, wherein the instructionsfurther cause the machine to calculate the first set of weights, thesecond set of weights, and the third set of weights in one processorcycle.
 13. The non-transitory computer readable medium as defined inclaim 8, wherein the instructions further cause the machine to calculatethe first weighted response, the second weighted response, and the thirdweighted response by bypassing multivariate reweighting to improve anoperating efficiency of the machine by calculating the first weightedresponse, the second weighted response, and the third weighted responseby applying maximum entropy to a constraint matrix.
 14. Thenon-transitory computer readable medium as defined in claim 8, whereinthe instructions further cause the machine to cause the display of thereport via a webpage in a first state with a set of options, the set ofoptions selectable by a user to change to the first state.
 15. Anapparatus comprising: means for calculating a weight to: calculate afirst set of weights for a first set of covariates corresponding to afirst treatment dataset using maximum entropy; calculate a second set ofweights for a second set of covariates corresponding to a secondtreatment dataset using maximum entropy; and calculate a third set ofweights for a third set of covariates corresponding to a control datasetusing maximum entropy, the weight calculating means to increase anoperational efficiency of the apparatus by calculating the first set ofweights, the second set of weights, and the third set of weightsindependently; means for determining a weighted response to: calculate afirst weighted response for the first treatment dataset, calculate asecond weighted response for the second treatment dataset, and calculatea third weighted response for the control dataset; and determine acausal effect between the first treatment dataset and the secondtreatment dataset based on a difference between the first weightedresponse and the second weighted response; and means for generating areport to display a report on a device of an audience measuremententity, the report including the causal effect, at least one of theweight calculating means, the weighted response determining means, orthe report generating means implemented by a logic circuit.
 16. Theapparatus as defined in claim 15, wherein the first treatment datasetcorresponds to a first advertisement campaign and the second treatmentdataset corresponds to a second advertisement campaign.
 17. Theapparatus as defined in claim 16, wherein the causal effect isindicative of a potential increase per individual exposed to the firstadvertisement campaign compared to the second advertisement campaign.18. (canceled)
 19. The apparatus as defined in claim 15, wherein theweight calculating means is to calculate the first set of weights, thesecond set of weights, and the third set of weights in one processorcycle.
 20. The apparatus as defined in claim 15, wherein the weightedresponse determining means is to calculate the first weighted response,the second weighted response, and the third weighted response bybypassing multivariate reweighting to improve an operating efficiency ofthe apparatus by calculating the first weighted response, the secondweighted response, and the third weighted response by applying maximumentropy to a constraint matrix.
 21. (canceled)
 22. An apparatuscomprising: memory; and at least one processor to execute machinereadable instructions to: calculate a first set of weights for a firstset of covariates corresponding to a first treatment dataset usingmaximum entropy; calculate a second set of weights for a second set ofcovariates corresponding to a second treatment dataset using maximumentropy; calculate a third set of weights for a third set of covariatescorresponding to a control dataset using maximum entropy, the at leastone processor to increase an operational efficiency of the apparatus bycalculating the first set of weights, the second set of weights, and thethird set of weights independently; calculate a first weighted responsefor the first treatment dataset, calculate a second weighted responsefor the second treatment dataset, and calculate a third weightedresponse for the control dataset; determine a causal effect between thefirst treatment dataset and the second treatment dataset based on adifference between the first weighted response and the second weightedresponse; and transmit a report to an audience measurement entity tocause display of the report on a device of the audience measuremententity.
 23. The apparatus as defined in claim 22, wherein the firsttreatment dataset corresponds to a first advertisement campaign and thesecond treatment dataset corresponds to a second advertisement campaign.24. The apparatus as defined in claim 23, wherein the causal effect isindicative of a potential increase per individual exposed to the firstadvertisement campaign compared to the second advertisement campaign.25. (canceled)
 26. The apparatus as defined in claim 22, wherein the atleast one processor is to calculate the first set of weights, the secondset of weights, and the third set of weights in one processor cycle. 27.The apparatus as defined in claim 22, wherein at least one processor isto calculate the first weighted response, the second weighted response,and the third weighted response by bypassing multivariate reweighting toimprove an operating efficiency of the apparatus by calculating thefirst weighted response, the second weighted response, and the thirdweighted response by applying maximum entropy to a constraint matrix.28. The apparatus as defined in claim 22, wherein the at least oneprocessor is to cause the display of the report via a webpage in a firststate with a set of options, the set of options selectable by a user tochange to the first state.